A353619 Side a of primitive integer-sided triangles (a, b, c) whose angle B = 3*C.

3, 35, 119, 112, 279, 20, 253, 539, 552, 91, 923, 533, 476, 1455, 224, 1504, 17, 799, 2159, 1513, 1476, 437, 1387, 3059, 2261, 1240, 3160, 4179, 2163, 748, 3212, 391, 1817, 5543, 3151, 4393, 5712, 1175, 2825, 7175, 5825, 2548, 5876, 189, 9099, 4077, 5859, 1736, 9352, 5768, 1189

Offset

1,1

Comments

The triples (a, b, c) are displayed in increasing order of side b, and if sides b coincide then in increasing order of the side c; hence, this sequence of sides a is not increasing.

In the case B = 3*C, the corresponding metric relation between sides is c*a^2= (b-c)^2 * (b+c).

Equivalently, length of side common to the two angles, one being the triple of the other, of a primitive integer-sided triangle.

For the corresponding primitive triples and miscellaneous properties and references, see A353618.

Links

Table of n, a(n) for n=1..51.

The IMO Compendium, Problem 1, 46th Czech and Slovak Mathematical Olympiad 1997.

Formula

a(n) = A353618(n, 1).

Example

According to inequalities between a, b, c, there exist 3 types of such triangles:
a < c < b with the smallest side a = 3 of the first triple (3, 10, 8).
c < a < b with the middle side a = 35 of the 2nd triple (35, 48, 27).
c < b < a with the largest side a = 539 of the 8th triple (539, 510, 216), the first of this type.

Maple

for b from 4 to 9000  do
  for q from 2 to floor((b-1)^(1/3)) do
a := (b-q^3) * sqrt(1+b/q^3);
if a= floor(a) and q^3 < b and igcd(a, b, q)=1 and (b-q^3) < a and a < b+q^3 then print(a); end if;
end do;
end do;

Crossrefs

Cf. A353618 (triples), this sequence (side a), A353620 (side b), A353621 (side c), A353622 (perimeter).

Cf. A343064 (similar, but with B = 2*C).

Sequence in context: A231645 A076376 A338787 * A133710 A130061 A368391

Adjacent sequences: A353616 A353617 A353618 * A353620 A353621 A353622

Keyword

nonn

Author

Bernard Schott, May 07 2022

Status

approved